Zernike polynomials: a guide

Lakshminarayanan, Vasudevan; Fleck, Andre

doi:10.1080/09500340.2011.633763

Cited by 72 publications

(63 citation statements)

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“…This technique is similar to the microspot LEED technique discussed in [20] but can also be used in PEEM mode. The Zernike polynomials ( , ) are a commonly used basis for the fitting of optical aberrations and the Zernike coefficients provide quantitative information about the low order geometric aberrations of the tested optics [21,22]. In case of PEEM-3, a 10-μm-size back focal plane aperture was scanned across the electron beam, allowing a narrow ray of electrons to pass to the detector.…”

Section: Hartmann Measurement Of Aberration Coefficientsmentioning

confidence: 99%

Hartmann characterization of the PEEM-3 aberration-corrected X-ray photoemission electron microscope

et al. 2018

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Aberration correction by an electron mirror dramatically improves the spatial resolution and transmission of photoemission electron microscopes. We will review the performance of the recently installed aberration corrector of the X-ray Photoemission Electron Microscope PEEM-3 and show a large improvement in the efficiency of the electron optics. Hartmann testing is introduced as a quantitative method to measure the geometrical aberrations of a cathode lens electron microscope. We find that aberration correction leads to an order of magnitude reduction of the spherical aberrations, suggesting that a spatial resolution of below 100 nm is possible at 100% transmission of the optics when using x-rays. We demonstrate this improved performance by imaging test patterns employing element and magnetic contrast.

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Section: Hartmann Measurement Of Aberration Coefficientsmentioning

confidence: 99%

Hartmann characterization of the PEEM-3 aberration-corrected X-ray photoemission electron microscope

et al. 2018

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“…The Zernike circle polynomials are an orthonormal basis set of functions whose general form in polar coordinates r r; θ is given by [17][18][19][20][21] Z n;m r N n;m R n;m rA m θ;…”

Section: B Zernike Polynomial Basismentioning

confidence: 99%

“…Figure 2 shows the shapes of several Zernike modes of different radial and angular orders. Surface plots for other modes can be found in [19].…”

Section: B Zernike Polynomial Basismentioning

confidence: 99%

“…One of those bases is formed by the well-known Zernike circle polynomials, whose applications in optics and astronomy have been thoroughly explored since their introduction by Zernike in the first-half of the XX century [17][18][19][20][21]. The main purpose of this work is to show the advantages of using Zernike polynomial series to analyze the measured all-sky night brightness distributions.…”

Section: Introductionmentioning

confidence: 99%

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Zernike analysis of all-sky night brightness maps

et al. 2014

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All-sky night brightness maps (calibrated images of the night sky with hemispherical field-of-view (FOV) taken at standard photometric bands) provide useful data to assess the light pollution levels at any ground site. We show that these maps can be efficiently described and analyzed using Zernike circle polynomials. The relevant image information can be compressed into a low-dimensional coefficients vector, giving an analytical expression for the sky brightness and alleviating the effects of noise. Moreover, the Zernike expansions allow us to quantify in a straightforward way the average and zenithal sky brightness and its variation across the FOV, providing a convenient framework to study the time course of these magnitudes. We apply this framework to analyze the results of a one-year campaign of night sky brightness measurements made at the UCM observatory in Madrid.

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“…Wavefront engineering -Genetic algorithm optimization enables the engineering of the wavefront to target a given Zernike polynomial 45,46 . Zernike polynomials > • , where stands for the radial degree and the azimuthal degree, are a set of orthonormal functions which are widely used in controlling and characterising the aberrations of optical systems.…”

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confidence: 99%

Tunable and free-form planar optics

et al. 2019

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The advent of spatial control over the phase and amplitude of light waves has profoundly transformed photonics, enabling major advances from imaging and information technology to biomedical optics. Here, we propose a novel approach to deterministic phase-front shaping through a planar thermo-optical module using designed microheaters to locally shape the refractive index distribution. When combined with a genetic algorithm optimisation, this SmartLens can produce free-form optical wavefront modifications. Individually or in arrays, it can generate complex functions based on either pure, or a combination of, Zernike polynomials, including lenses or aberration correctors of electrically-tuneable magnitude. This simple and compact concept complements the existing optical shaping toolbox by offering low chromatic aberrations, polarisation-insensitive and transmission-mode components which can be readily integrated into existing optical systems.

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Zernike polynomials: a guide

Cited by 72 publications

References 0 publications

Hartmann characterization of the PEEM-3 aberration-corrected X-ray photoemission electron microscope

Hartmann characterization of the PEEM-3 aberration-corrected X-ray photoemission electron microscope

Zernike analysis of all-sky night brightness maps

Tunable and free-form planar optics

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